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%I #16 Jul 17 2021 06:54:15
%S 1179,1274,1895,4775,5304,5874,6525,6639,13035,16380,17424,18459,
%T 21239,21584,21714,22475,22715,22734,27410,28304,29340,29909,31755,
%U 32294,34700,37700,41525,42164,42929,42950,43275,46415,47174,47300,53364,57879,59739,61194
%N Numbers k such that 8k+1, 12k+1 and 24k+1 are primes and the last two are also of the form x^2 + 27y^2, so the tetrahedral number T(24k+1) is a Fermat pseudoprime to base 2.
%C The first 3 terms were found by Rotkiewicz.
%C The generated tetrahedral pseudoprimes are 3776730328549, 4765143438329, 15680770945781, ...
%H Amiram Eldar, <a href="/A321867/b321867.txt">Table of n, a(n) for n = 1..10000</a>
%H Andrzej Rotkiewicz, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21137.pdf">On some problems of W. Sierpinski</a>, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
%e 1179 is in the sequence since 8*1179+1 = 9433, 12*1179+1 = 14149 = 107^2 + 27*10^2 and 24*1179+1 = 28297 = 163^2 + 27*8^2 are primes.
%t sqQ[n_] := n>0 && IntegerQ[Sqrt[n]]; sqsumQ[n_] := PrimeQ[n] && False =!= Reduce[ x^2 + 27 y^2 == n, {x, y}, Integers]; aQ[n_] := PrimeQ[8n+1] && sqsumQ[12n+1] && sqsumQ[24n+1]; Select[Range[100000], aQ]
%Y Cf. A000292, A001567, A014752, A321866.
%K nonn
%O 1,1
%A _Amiram Eldar_, Nov 20 2018
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